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Sin pi/3 __ __ pi/6 = 1/2(sin pi/2 + sin pi/6)

I think I’m just supposed to fill in the blank? (question off of a p e x) please give explanation!

Sin pi/3 __ __ pi/6 = 1/2(sin pi/2 + sin pi/6) I think I’m just supposed to fill in-example-1
User Milanseitler
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1 Answer

11 votes
11 votes

Notice that

π/2 = π/3 + π/6

π/6 = π/3 - π/6

Recall the angle sum identities for sine:

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

sin(x - y) = sin(x) cos(y) - cos(x) sin(y)

By adding these together, we get

sin(x + y) + sin(x - y) = 2 sin(x) cos(y)

==> sin(x) cos(y) = 1/2 (sin(x + y) + sin(x - y))

Now take x = π/3 and y = π/6 :

sin(π/3) cos(π/6) = 1/2 (sin(π/2) + sin(π/6))

So the blank should be filled with cos.